Book VIII · Proposition 11

VIII.11

Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate of that which the side has to the side.Heath, 1908

One mean proportional between two squares, and the ratio is duplicate. The arithmetical twin of VI.19.

Every step, checked

• the sides are genuine numbersgiven
• the number found is a mean proportionalby VIII.11
• and it is the only oneby VIII.8
• so square is to square in the duplicate ratio of side to sideby VIII.11

What it needs, and what needs it

Needs: VIII.8

Used by: VIII.18

Depth: 2 steps of argument above the first principles. Parallel postulate: not needed.

What it takes on trust

Nothing. It draws no intersections and reads nothing off the picture.

The proposition as code

@proposition(
    "VIII.11",
    THEOREM,
    sample=lambda rng: (rng.randint(2, 20), rng.randint(2, 20)),
    note="One mean proportional between two squares, and the ratio is duplicate. "
    "The arithmetical twin of VI.19.",
)
def prop_VIII_11(a: int, b: int) -> Out:
    hypothesis("the sides are genuine numbers", a > 1 and b > 1, guard=True)
    first, second = a * a, b * b
    mean = a * b
    because(prop_VIII_8, a, b, 3)

    claim("the number found is a mean proportional", "VIII.11",
          first * second == mean * mean)
    claim("and it is the only one", "VIII.8",
          not any(first * second == m * m for m in range(1, mean)))
    claim("so square is to square in the duplicate ratio of side to side", "VIII.11",
          first * (b * b) == second * (a * a))
    return Out(mean=mean)